Exercise 9, § 2

a) Let $X$ be a countable linearly ordered set without gaps (Exercise 7). Show that there is a strictly increasing bijective mapping $\varphi$ of $X$ onto one of the four intervals of $Q$ with end-points 0 and 1. [Arrange the elements of $X$ and the points of the appropriate interval of $Q$ in sequences $(a_n), (b_n)$, and define $\varphi$ by induction.] If $X$ is endowed with the topology $\mathfrak{T}_0(X)$, then $\varphi$ is a homeomorphism of $X$ onto $\varphi(X)$.

b) Deduce from a) that every countable dense subset of an open interval ]$a, b[$ of $R$ is homeomorphic to $Q$.

c) Deduce from a) that if $X$ is any countable ordered set endowed with the topology $\mathfrak{T}_0(X)$, then there exists a strictly increasing homeomorphism of $X$ onto a subspace of $Q$ [embed $X$ in a countable linearly ordered space $X'$ without gaps, such that $\mathfrak{T}_0(X)$ is induced by $\mathfrak{T}_0(X')$].